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I am looking to understand the Newman-Penrose formalism. I understand that the spin coefficients are determined by a tetrad $ \{ e_{1}^{\alpha},e_{2}^{\alpha},e_{3}^{\alpha},e_{4}^{\alpha} \}$ = $\{l^{\alpha}, n^{\alpha}, m^{\alpha}, \bar{m}^{\alpha} \}$ where the overbar denotes a complex conjugate.

The spin coefficients are determined in terms of Ricci coefficients:

$$ \gamma_{abc} = g_{\mu \lambda} e^{\mu}_{a} e^{\nu}_{c} \nabla_{\nu} e_{b}^{\lambda} $$Suppose I take one of the spin coefficients,

$$ \pi = -\gamma_{241}$$

To get the complex conjugate of this coefficient, can I just switch all the 3's to 4s and vice versa, i.e.

$$ \bar{\pi} = -\gamma_{231} $$?

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Yes. See e.g. Chandrasekhar, The Mathematical Theory of Black Holes, pg. 42.

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